A torsion-only extension of the canonical formula fails for Frobenius

The morphism, its degree and its ramification. The coordinate function $x$ has ${\mathrm{ord}}_{\mathrm{\infty }}(x)=-1$ by [F1], so $x^p$ is nonconstant, and [F2] gives the finite surjective morphism $\phi$ of degree $[k({\mathbb{P}}_k^1):k(x^p)]=p$, since $1,x,\dots ,x^{p-1}$ is a basis over $k(x^p)$. On the two charts the maps are $k[x]\to k[x]$, $x\mapsto x^p$, and $k[y]\to k[y]$, $y\mapsto y^p$, because $y=x^{-1}$. The points $0$ and $\mathrm{\infty }$ are $k$-rational with uniformizers $x$ and $y$ by [F1], so [F3] gives $e_0={\mathrm{ord}}_0(x^p)=p$ and $e_{\mathrm{\infty }}={\mathrm{ord}}_{\mathrm{\infty }}(y^p)=p$. The zero and pole fibres are supported respectively at $0$ and $\mathrm{\infty }$; the fibre degree formula [F2] at $0$ is consistent with $\mathrm{deg}(\phi )=e_0[\kappa (0):\kappa (0)]=p$.

Ramification after geometric base change. Over $\bar{k}$, let $a$ be any finite target point and choose $b\in \bar{k}$ with $b^p=a$. The pullback of the target parameter $x-a$ is $x^p-a=(x-b{)}^p$, so the index at the geometric point $b$ is $p$. On the infinity chart the same calculation is $y\mapsto y^p$. Thus every geometric closed point is index-ramified. Over an imperfect original field the indices of its closed points need not all be $p$: for example, if $k={\mathbb{F}}_p(a)$ with $a$ not a $p$-th power, the target point $t=a$ has preimage defined by the irreducible polynomial $x^p-a$. Its local uniformizer is $x^p-a$, exactly the pullback of $t-a$, so its index is $1$, while the residue extension is purely inseparable.

The pullback of differentials is the zero map. The canonical bundle of the target is generated on $U_0$ by $\mathrm{d}x$, and the pullback map ${\phi }^{\ast }{\omega }_{{\mathbb{P}}_k^1}\to {\omega }_{{\mathbb{P}}_k^1}$ sends its generator ${\phi }^{\ast }(\mathrm{d}x)$ to $\mathrm{d}(x^p)=p{x}^{p-1}\mathrm{d}x=0$, because $p=0$ in $k$. The same computation in coordinate $y$ on $U_{\mathrm{\infty }}$ gives ${\phi }^{\ast }(\mathrm{d}y)\mapsto \mathrm{d}(y^p)=0$. By the right-exact transitivity sequence [F6], the map ${\phi }^{\ast }{\omega }_{{\mathbb{P}}_k^1}\to {\omega }_{{\mathbb{P}}_k^1}$ is followed by the quotient to ${\mathrm{\Omega }}_{C\mathrm{/}D}$. Since the first map is zero, this quotient is an isomorphism. Thus the pullback of differentials is not injective; the injectivity assertion in the canonical-bundle theorem [F5] is unavailable because its separability hypothesis fails.

The relative differentials are invertible, with no torsion. On the source chart $U_0$ the map is $A=k[x]\to B=k[z]$, $x\mapsto z^p$, so $B=A[T]\mathrm{/}(T^p-x)$ with $T\mapsto z$; the derivative of $g(T)=T^p-x$ is $g^{\mathrm{\prime }}(T)=p{T}^{p-1}=0$, and [F4] gives ${\mathrm{\Omega }}_{B\mathrm{/}A}\cong B\mathrm{/}(g^{\mathrm{\prime }})=B$, free of rank one on $\mathrm{d}T$. The same holds on $U_{\mathrm{\infty }}$ in coordinate $y$. Thus the relative differential sheaf for $\phi$, ${\mathrm{\Omega }}_{C\mathrm{/}D}$, is invertible, so its torsion subsheaf is zero. Set $l_p^{\mathrm{t}\mathrm{o}\mathrm{r}}:={\mathrm{length}}_{{\mathcal{O}}_{C,p}}(\mathrm{tors}({\mathrm{\Omega }}_{C\mathrm{/}D}{)}_p)$; then $l_p^{\mathrm{t}\mathrm{o}\mathrm{r}}=0$ for every closed point $p$.

The recipe gives $R_f^{\mathrm{t}\mathrm{o}\mathrm{r}}=0$, but the isomorphism fails. By step 2.2 all coefficients $l_p^{\mathrm{t}\mathrm{o}\mathrm{r}}$ vanish. The canonical divisor computation is $\mathrm{div}(\mathrm{d}x)=-2[\mathrm{\infty }]$: $\mathrm{d}x$ is a frame on $U_0$, and $\mathrm{d}x=\mathrm{d}(y^{-1})=-y^{-2}\mathrm{d}y$ on $U_{\mathrm{\infty }}$. Thus ${\omega }_{{\mathbb{P}}_k^1}\cong {\mathcal{O}}_{{\mathbb{P}}_k^1}(-2[\mathrm{\infty }])$. The pullback divisor is ${\phi }^{\ast }[\mathrm{\infty }]=p[\mathrm{\infty }]$, since its fibre is supported at $\mathrm{\infty }$ with index $p$ from step 1.1; [F7] gives ${\phi }^{\ast }{\omega }_{{\mathbb{P}}_k^1}\cong {\mathcal{O}}_{{\mathbb{P}}_k^1}(-2p[\mathrm{\infty }])$. If the proposed formula held with $R_f^{\mathrm{t}\mathrm{o}\mathrm{r}}=0$, these divisor line bundles would be isomorphic, so $(2p-2)[\mathrm{\infty }]$ would be principal by [F7]. Its degree is $2p-2>0$, contradicting [F7], which gives degree zero for principal divisors.

Conclusion. The $p$-th-power map is finite surjective of degree $p$; its indices at $0$ and $\mathrm{\infty }$ are $p$, and after base change every geometric closed point has index $p$. Its relative differentials are invertible with zero torsion, so the proposed torsion-submodule recipe gives $R_f^{\mathrm{t}\mathrm{o}\mathrm{r}}=0$. The formula fails because ${\omega }_{{\mathbb{P}}_k^1}\cong \mathcal{O}(-2[\mathrm{\infty }])$ and ${\phi }^{\ast }{\omega }_{{\mathbb{P}}_k^1}\cong \mathcal{O}(-2p[\mathrm{\infty }])$ are not isomorphic. Thus separability cannot be dropped when extending the formula by this recipe, and this conclusion does not define the separable different divisor for an inseparable map.